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Journal of Physics A Mathematical and Theoretical

Generalized coherent, squeezed and intelligent states for exactly solvable quantum systems and the analogue of the displacement and squeezing operators

A N F Aleixo, A. B. Balantekin

Published Jul 12, 2013• 8 citations
Publisher

Abstract

Using an algebraic approach, we construct purely coherent and squeezed states for a set of exactly solvable quantum confined systems assuming a generalization of both the ladder-operator and the displacement-operator methods, showing the equivalence between the two procedures. We obtain the analogue of the displacement and the squeeze operators and study their basic relations and their connections with the properties of the purely coherent and squeezed states that they produce. In a second approach, we construct generalized squeezed coherent and intelligent states for these systems considering the successive action of the displacement and the squeeze operators (in the first case) and the combined action of these operators (in the second case). We apply the generalized approach for shape-invariant systems with potential parameters related by a translation and by a scaling and show that the presence of the generalizing functionals of the potential parameters in the formalism opens a large number of possibilities in the construction of these special quantum states.

DOI
10.1088/1751-8113/46/31/315303

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Using an algebraic approach, we construct purely coherent and squeezed states for a set of exactly solvable quantum confined systems assuming a generalization of both the ladder-operator and the displacement-operator methods, showing the equivalence between the two procedures. We obtain the analogue of the displacement and the squeeze operators and study their basic relations and their connections with the properties of the purely coherent and squeezed states that they produce. In a second approach, we construct generalized squeezed coherent and intelligent states for these systems considering the successive action of the displacement and the squeeze operators (in the first case) and the combined action of these operators (in the second case). We apply the generalized approach for shape-invariant systems with potential parameters related by a translation and by a scaling and show that the presence of the generalizing functionals of the potential parameters in the formalism opens a large number of possibilities in the construction of these special quantum states.
Using an expanded algebraic formalism with the inclusion of inverse operators, we construct raised and decreased coherent states for a set of exactly solvable quantum confined systems. We assume in this procedure both the ladder-operator and the displacement-operator methods, showing the equivalence between the two approaches. For each coherent state defined, we present its expansion in the Hilbert eigenstate space Hes, eigenvalue equation, overcompleteness relation, as well as other intrinsic properties. Whenever possible, we present an interpretation based on nonlinear deformation models for these new forms of coherent states. We evaluate the relevance of the new coherent states in quantum entanglement and squeezing by taking, as an example, the case of a coupled system.
We consider quantum confined systems represented by supersymmetric quantum mechanics where the Hamiltonians are products of two operators. We introduce inverses of these operators as well as pair operators, which are products of the original operators and their inverses. We show that the eigenstates of these pair operators are purely squeezed states, and using them we construct raised and decreased extensions of the squeezed states. We apply this procedure both to the ladder-operator and to the displacement-operator methods, showing the equivalence between those the two approaches. Finally, we show how the raised and decreased squeezed states can be dynamically produced by coupling the Hamiltonian with a two-level atom.
The generalized purely squeezed states for primary shape-invariant potentials systems, quantum deformed by different models, are constructed by the ladder-operator method within an algebraic approach based on supersymmetric quantum mechanics. The characteristic properties of these states as well as their quantum statistical properties and squeezing effects for generalized quadrature observables are studied and analyzed in terms of the quantum deformation parameter q. An application is given for a quantum deformed Pöschl–Teller potential system, and numerical results are presented and discussed in detail.
We obtain the Robertson–Schrödinger uncertainty relation for shape-invariant systems and construct generalized Robertson intelligent states for these systems using an algebraic approach based on the supersymmetric quantum mechanics. Using the variances of generalized quadrature operators we study the coherency and squeezing properties, evaluating their dependence on different kinds of generalizations for shape-invariant systems with potential parameters related by a translation (Pöschl–Teller potential) and by a scaling (self-similar potential).
Among the three methods of defining the squeezed states of the harmonic oscillator, only two, i.e. the minimum uncertainty and ladder-operator methods were extended to investigate the squeezed states of the general symmetry systems. The only exception is the displacement-operator formalism. In this letter we propose a displacement-operator approach to the squeezed states for general systems. We show that the ladder-operator squeezed states for general symmetry systems can be obtained by applying an exponential displacement-operator on the ``vacuum'' state. The general formalism is presented and it is illustrated with su(1,1) as an explicit example. As a byproduct, the squeezed states of the q-deformed oscillator are obtained and, in particular, expressed in terms of the {\\it usual exponential} displacement-operator acting the ``vacuum'' state.